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There's a Jazz Jam at Churchill's Pub on Monday nights that's great, that's in wynwood not too far from the beach. (You can catch some really crazy stuff there on other nights, never covers)
The Bar in coral Gables has cover bands on Friday or Saturday nights, you'll have to look it up.
On South Beach there is a bar called Love Hate that used to have music, and I've played at a place called Karu & Y or something to that effect. There's also Jazzid what used to have a range of local musicians playing.
Other locations to check out in downtown (also not far from the beach) are Electric Pickle, Black Bird Ordinary and The Stage (Actually that might be your best bet but check their schedule)

Bayside (in downtown) should have a cover band on the weekends

If you can get north about half an hour there is a great venue that never fails to book good music called both Kreepy Tiki & Monterey Club

Local groups that are popular and fun to watch are mostly Hispanic music. Suenalo, Spam Allstars and Bachacco. There's really a lot of electronic music around here. Some that I actually enjoy are Afrobeta and Beatmachines.